Rapid Mixing from Spectral Independence beyond the Boolean Domain

Feng, Weiming; Guo, Heng; Yin, Yitong; Zhang, Chihao

doi:10.1137/1.9781611976465.95

Cited by 30 publications

(26 citation statements)

References 46 publications

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“…v } be the collection of feasible vertex-spin pairs under τ , where Ω τ v represents the set of feasible spins at v conditioned on τ . The following definition is taken from [CG ŠV21]; see also [ALO20,FGYZ21].…”

Section: Spectral Independence Via Stability Of the Partition Functionmentioning

confidence: 99%

“…Spectral independence serves as an important technical bridge: for many results established with different approaches, their proofs also imply spectral independence and as a consequence we obtain much stronger running time guarantees. In particular, in [CLV20] it was shown that potential function techniques used for analyzing correlation decay algorithms also imply spectral independence; see also [CG ŠV21,FGYZ21]. Very recent works [BCC + 21, Liu21] prove that path coupling arguments for a broad class of chains, including any local chain, yields spectral independence.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Independence via Stability and Applications to Holant-Type Problems

Chen¹,

Liu²,

Vigoda³

2021

Preprint

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This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point λ then spectral independence holds at λ. As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal O(n log n) mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvinok (2017), refined by Patel and Regts (2017).There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an O(n log n) sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an O(n log n) sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.

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Section: Spectral Independence Via Stability Of the Partition Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spectral Independence via Stability and Applications to Holant-Type Problems

Chen¹,

Liu²,

Vigoda³

2021

Preprint

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“…Then for some absolute constant ǫ ≈ 10 −5 , if q ≥ ( 11 6 − ǫ)∆ , then the uniform distribution over proper list-colorings for (G, L) is ℓ ∞ -independent with parameter O(1). Theorem 3.6 (See proof of Theorem 9 [Che+21], see also Lemma 6.1 [Fen+21]). Let ǫ > 0, and suppose that (G, L) is a list colouring instance where G = (V, E) is a triangle-free graph of maximum degree ∆ and L = (L(v)) v∈V is a collection of color lists of maximum length q ≥ (1 + ǫ)α * ∆ + 1, α * ≈ 1.763.…”

Section: Gibbs Distributions Of Spin Systemsmentioning

confidence: 98%

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Kaufman¹,

Kyng²,

Soldà³

2021

Preprint

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We show new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over {0, 1} n . Building on developments in high-dimensional expanders (Kaufman and Mass ITCS'17, Dinur and Kaufman FOCS'17, Kaufman and Oppenheim Combinatorica'20) and matroid theory (Adiprasito et al. Ann. Math.'18), a breakthrough result of Anari, Liu, Oveis and Vinzant (STOC '19) showed that the up-down random walk on matroid bases has polynomial mixing time -making it possible to efficiently sample from the (weighted) uniform distribution over matroid bases. Since then, there has been a flurry of related work proving polynomial mixing times for random walks used to sample from a wide range of discrete probability distributions. Many works have observed that as a corollary of their mixing time analysis, one can obtain scalar concentration for 1-Lipschitz functions of samples from the stable distribution via standard arguments that convert bounds on the modified log-Sobolev (MLS) or Poincaré constant of a random walk into concentration results for the associated stable distribution of the random walk, see for example Hermon and Salez (arXiv'19). Several recent works have considered a matrix analog of the Poincaré inequality for a random walk with an associated stable distribution. Using this matrix Poincaré inequality, these works have derived a concentration result for matrix-valued spectral norm-Lipschitz functions of samples from the distribution, see for example Auon et al. (Adv. Math.'20). Unfortunately, these bounds are weak in many important regimes.A recently developed strategy for analyzing up-down walks is based around a novel notion of spectral independence, which quantifies the dependence between variables in a distribution over {0, 1} n using the largest eigenvalue of an associated pairwise influence matrix I, see Anari et al. (FOCS'20). Many works on spectral independence have in fact bounded a stronger quantity I ∞→∞ ≥ λ max (I), which we call ℓ ∞ -independence. We show that any distribution over {0, 1} n which has bounded ℓ ∞ -independence satisfies a matrix Chernoff bound that in key regimes is much stronger than spectral norm-Lipschitz function concentration bounds derived from matrix Poincaré inequalities. Our bounds match the matrix Chernoff bound for independent random variables due to Tropp, which is the strongest known in many cases and is essentially tight for several key settings. For spectral graph sparsification, our matrix concentration results are exponentially stronger than those obtained from matrix Poincaré inequalities. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18). Using our bound, we can conclude as a corollary that a union of O(log |V |) random spanning trees gives a spectral graph sparsifier of a graph with |V | vertices with high probability, matching results for independent edge sampling, and matching lower bounds from Kyng and Song. This improves on the O(log 2 |V |) spanning tr...

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“…A distribution over {−1, +1} is said to be -spectrally independent with all fields if max (Ψ inf ( ) ) ≤ for any = ( ) ∈ ∈ R >0 . The notion of spectral independence was first introduced in [ALO20], then further developed in [CGŠV21,FGYZ21]. Remark that if a distribution is -spectrally independent in ∞-norm with all fields, then it must be -spectrally independent with all fields.…”

Section: F L C U B Fmentioning

confidence: 99%

Optimal Mixing Time for the Ising Model in the Uniqueness Regime

Chen¹,

Feng²,

Yin³

et al. 2021

Preprint

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A. We prove an optimal ( log ) mixing time of the Glauber dynamics for the Ising models with edge activity ∈ Δ−2 Δ , Δ Δ−2 . This mixing time bound holds even if the maximum degree Δ is unbounded. We refine the boosting technique developed in [CFYZ21], and prove a new boosting theorem by utilizing the entropic independence defined in [AJK + 21]. The theorem relates the modified log-Sobolev (MLS) constant of the Glauber dynamics for a near-critical Ising model to that for an Ising model in a sub-critical regime.

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Rapid Mixing from Spectral Independence beyond the Boolean Domain

Cited by 30 publications

References 46 publications

Spectral Independence via Stability and Applications to Holant-Type Problems

Spectral Independence via Stability and Applications to Holant-Type Problems

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Optimal Mixing Time for the Ising Model in the Uniqueness Regime

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